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Formal manifolds: local structure of morphisms, and formal submanifolds

Fulin Chen, Binyong Sun, Chuyun Wang

TL;DR

The paper advances the theory of formal manifolds by establishing a robust local-structure framework for morphisms, including a formal inverse-function type result and a comprehensive constant-rank theorem that yields canonical local models. It introduces and analyzes formal submanifolds, defining immersive and embedded notions via surjectivity on structure sheaves, and proves a universal property for initial immersed formal submanifolds, together with level-set Cartesian diagrams. The work unifies and extends ideas from supergeometry, providing a structured approach to reductions, tangent spaces, differentials, and local coordinates in the formal setting, with potential applications to smooth relative Lie algebra (co)homologies and formal Lie groups. Overall, the results give precise local models and universal constructions that link formal and smooth submanifolds, enabling Cartesiantype descriptions of level sets and submanifold embeddings in the formal context.

Abstract

This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In three previous papers, we introduce the notion of formal manifolds and study their basic theory, focusing on function spaces and Poincare's lemma. In this paper, we further explore the foundational framework of formal manifolds, including the local structure of constant rank morphisms (such as inverse function theorem and constant rank theorems) as well as the theory of formal submanifolds.

Formal manifolds: local structure of morphisms, and formal submanifolds

TL;DR

The paper advances the theory of formal manifolds by establishing a robust local-structure framework for morphisms, including a formal inverse-function type result and a comprehensive constant-rank theorem that yields canonical local models. It introduces and analyzes formal submanifolds, defining immersive and embedded notions via surjectivity on structure sheaves, and proves a universal property for initial immersed formal submanifolds, together with level-set Cartesian diagrams. The work unifies and extends ideas from supergeometry, providing a structured approach to reductions, tangent spaces, differentials, and local coordinates in the formal setting, with potential applications to smooth relative Lie algebra (co)homologies and formal Lie groups. Overall, the results give precise local models and universal constructions that link formal and smooth submanifolds, enabling Cartesiantype descriptions of level sets and submanifold embeddings in the formal context.

Abstract

This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In three previous papers, we introduce the notion of formal manifolds and study their basic theory, focusing on function spaces and Poincare's lemma. In this paper, we further explore the foundational framework of formal manifolds, including the local structure of constant rank morphisms (such as inverse function theorem and constant rank theorems) as well as the theory of formal submanifolds.
Paper Structure (19 sections, 50 theorems, 316 equations)

This paper contains 19 sections, 50 theorems, 316 equations.

Table of Contents

  1. Introduction and main results
  2. Basics on formal manifolds
  3. Local structure of morphisms
  4. Formal submanifolds
  5. Tangent spaces, differentials, and inverse function theorem
  6. Tangent spaces
  7. Differentials
  8. Proof of Theorem \ref{['thm:invfun']}
  9. Constant rank theorems
  10. Morphisms to $N^{(k)}$
  11. Constant rank morphisms
  12. Proof of Theorem \ref{['thm:constantrankthmmain']}
  13. Immersions and submersions
  14. Formal submanifolds
  15. Proof of Theorem \ref{['thm:charclosedsub']}
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $\varphi=({\overline\varphi}, \varphi^*): (M',{\mathcal{O}}')\rightarrow (M,{\mathcal{O}})$ be a morphism of formal manifolds, and let $b\in M'$. Assume that are both bijective. Then there exists an open neighborhood $U'$ of $b$ in $M'$ such that $U:=\overline\varphi(U')$ is an open subset of $M$, and the restriction of $\varphi$ on $U'$ is an isomorphism of formal manifolds.

Theorems & Definitions (107)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Example 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.11
  • ...and 97 more